A non-nested unstructured mesh perspective on highly parallel multilevel smoothed Schwarz preconditioner for linear parametric PDEs

The multilevel Schwarz preconditioner is one of the most popular parallel preconditioners for enhancing convergence and improving parallel efficiency. However, its parallel implementation on arbitrary unstructured triangular/tetrahedral meshes remains challenging. The challenges mainly arise from the inability to ensure that mesh hierarchies are nested, which complicates parallelization efforts. This paper systematically investigates the non-nested unstructured case of parallel multilevel algorithms and develops a highly parallel non-nested multilevel smoothed Schwarz preconditioner. The proposed multilevel preconditioner incorporates two key techniques. The first is a new parallel coarsening algorithm that preserves the geometric features of the computational domain. The second is a corresponding parallel non-nested interpolation method designed for non-nested mesh hierarchies. This new preconditioner is applied to a broad range of linear parametric problems, benefiting from the reusability of the same coarse mesh hierarchy for problems with different parameters. Several numerical experiments validate the outstanding convergence and parallel efficiency of the proposed preconditioner, demonstrating effective scalability up to 1,000 processors.